Triangles dominate GRE geometry. Make sure you learn the area formula for a triangle (½ base × height) plus facts like “the sum of a triangle’s interior angles is 180°.” But you probably already know about that stuff. Here are some GRE triangle facts you may not know about.
The height of a triangle can be measured on the outside.
To find a triangle’s height, you measure the length of a line segment that starts at one corner—a vertex—and meets the opposite side—the base—at a 90º angle.
You’re probably used to drawing a height line on the inside of a triangle that runs from the top corner to the bottom side. But what if that line doesn’t meet the bottom of the triangle at a right angle? You can draw the height line on the outside so that it’s perpendicular to the the plane of the bottom side.
What else you can do inside qs leap ?
What else you can do inside qs leap ?
The base of a triangle can be any side.
Or you can pick a side other than the bottom to be the base. Although “bottom” and “base” are synonyms in everyday speech, in geometry jargon a triangle’s “base” is whichever side you want. So, if you prefer, pick a side that places the height line inside the triangle.
The height of an equilateral triangle is one-half the side length times the square root of three (½ s√3).
For a triangle with equal sides, finding the height is easy. You just need to know the side length and these two basic triangle facts.
- An equilateral triangle can be split into two equal 30-60-90 triangles.
- A 30-60-90 triangle has sides in a ratio of a : a√3 : 2a, where a is the shortest leg and 2a is the hypotenuse.
In an equilateral triangle, the side length s equals 2a, and the height h equals a√3, so the height works out to ½ s√3.
- h = a√3
- 2a = s
- a = ½ s
- h = ½ s√3
You don’t need to memorize this proof, of course. It just shows you why the one part you do need to remember—h = ½ s√3—will give you the height of an equilateral triangle once you know the side length.
The diagonal of a square is the side length times the square root of two (s√2).
Speaking of splitting shapes into right triangles, finding the diagonal of a square comes down to knowing its side length plus this pair of triangle facts.
- A square can be split into two equal 45-45-90 triangles.
- A 45-45-90 triangle has sides in a ratio of s : s : s√2, where s is the leg and s√2 is the hypotenuse.
The sides of a square are the same as the legs of the right triangles it contains, so a square’s diagonal d must be the hypotenuse shared by those two triangles. So make sure you memorize that d = s√2 for a square.
Right triangles on the GRE often have sides in a ratio of 3:4:5 or 5:12:13.
Side-length ratios for right triangles are a recurring theme in this post—and for good reason. GRE Quant favors not just triangles but often right triangles in particular. 30-60-90 and 45-45-90 triangles, both named for their angles, are two common types.
Two more are 3:4:5 and 5:12:13 right triangles, both named for the ratio of their side lengths. Each of these ratios forms what geometry junkies call a “Pythagorean Triple” or whole-number solution to the Pythagorean Theorem: a2 + b2 = c2.
Memorize 3:4:5 and 5:12:13 and, in many cases, you won’t need the Pythagorean Theorem. For instance, if you need to find c when a and b are 6 and 8, don’t use the theorem: recognize the ratio.
- 6 : 8 : c = 2(3) : 2(4) : 2(5)
- c = 2(5)
- c = 10
The legs 6 and 8 are twice 3 and 4, so the hypotenuse must be twice 5 and, thus, 10.
Triangles hide in other figures in GRE Quant.
You’ll see shapes besides triangles on the exam, such as circles, squares, and more. Yet even then a triangle may be lurking. For instance, if you get a question with a square inscribed in a circle, get ready to work with a 45-45-90 triangle. In fact, whenever you encounter right angles (in a square or elsewhere) in a problem, consider solutions that involve right triangles.
Photo: “ABC Museo Madrid“